Scattered Data Approximation by LR B-Spline Surfaces: A Study on Refinement Strategies for Efficient Approximation

Optimal Surface Fitting of Point Clouds Using Local Refinement

Skytt

Dokken

2022

Self Cite

With the development of high rate sensors based on LIDAR (light detection and ranging) and sonar technology, geospatial data representing terrain or seabed often contains millions of points. Performing a surface approximation of the point clouds is an elegant way to reduce noisy and unorganized data to a mathematical surface with just a few coefficients to estimate. Traditional spline surfaces are able to compactly represent smooth shapes, but lack the ability to adapt the representation locally to the point clouds. Locally Refined (LR) B-spline surfaces address that challenge as they have the nice property of being locally refinable. Their format can be made compatible with most Geographic Information System (GIS) software, and they facilitate various applications such as the drawing of contour lines or spatio-temporal deformation analysis. This introduction aims to explain the need for surface approximation, and present the state of the art in that domain. We compare the LR B-spline approach with different methods for surface approximation including raster, and triangular irregular networks.

Section: Discussionmentioning

confidence: 99%

Introduction

Optimal Surface Fitting of Point Clouds Using Local Refinement

Skytt

Dokken

2022

Self Cite

“…An extensive introduction of new degrees of freedom may be appropriate. We will focus on approximation accuracy related to data size as in [Sky22].…”

Section: Isogeometric Analysis Versus Scattered Data Approximationmentioning

confidence: 99%

“…3.1.1 and the iteration is pursued until the prescribed tolerance is met or no further accuracy improvements are possible. Here we give a short overview over the refinement strategies pursued in [Sky22], which are partly the same as in [Joh13]. The methods will be applied to various data sets in Chaps.…”

Section: Refinement Strategies In the Approximation Contextmentioning

confidence: 99%

“…The latter means that not all candidate mesh cells or B-splines are refined at each iteration level. The candidates are sorted according to selected criteria involving the number of points outside the tolerance belt and the residual value in these points [Sky22].…”

Section: Refinement Strategies In the Approximation Contextmentioning

confidence: 99%

“…Bilinear polynomials often results in lean surfaces, but the stability suffers for some data sets and some refinement strategies. We will build on the results from [Sky22] and complement them with statistical considerations in Chap. 4.…”

Section: Refinement Strategies In the Approximation Contextmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive Surface Fitting with Local Refinement: LR B-Spline Surfaces

Optimal Surface Fitting of Point Clouds Using Local Refinement

Skytt²,

Dokken³

2022

Self Cite

A locally refined (LR) B-spline surface is a piecewise polynomial surface for which the distribution of the surface coefficients can be locally adapted. Such a mathematical representation is interesting for fitting scattered and noisy data, as the local behaviour of a real point cloud may require more degrees of freedom only locally. The number of redundant surface coefficients is minimized, which avoids the fitting of the point cloud’s noise. The surface approximation is performed iteratively either by solving a least squares system or by a local approximation method. This procedure allows for mesh refinement in domains where the distance between a current surface and the point cloud exceeds a prescribed tolerance. In this way, parts of the LR B-spline surface obtained at previous steps may be kept unchanged. This chapter aims at explaining the adaptive fitting using local refinement with LR B-splines. We present two examples with simulated point clouds to illustrate the methodology.

A Statistical Criterion to Judge the Goodness of Fit of LR B-Splines Surface Approximation

Optimal Surface Fitting of Point Clouds Using Local Refinement

Skytt

Dokken

2022

Self Cite

The surface approximation obtained with adaptive strategies using locally refined (LR) B-splines depends on the degrees of freedom of the spline space, the tolerance from which the refinement is performed, the noise level of the scattered observations, the refinement strategy and the bidegree of the spline space. The choice of the best model is a challenging task that can be partially answered with statistical criteria, such as the Akaike Information Criterion (AIC). Here we relax the assumption that the approximation error should be normally distributed and with equal variance and propose the use of the student distribution to compute the AIC. We apply the AIC to decide which tolerance, refinement level, or polynomial bidegree are the most adequate for an optimal fitting. We highlight how the resulting AIC can be combined with more usual criteria to judge the goodness of fit of the surface approximation.